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A specialized high-precision numerical search for equal-mass collisionless three-body periodic free-fall orbits with central symmetry is conducted. The search is based on Newton's method with initial approximations obtained by the…
The paper addresses kinematic and geometrical aspects of the Orthoglide, a three-DOF parallel mechanism. This machine consists of three fixed linear joints, which are mounted orthogonally, three identical legs and a mobile platform, which…
We study the dynamics of $N$ point vortices on a rotating sphere. The Hamiltonian system becomes infinite dimensional due to the non-uniform background vorticity coming from the Coriolis force. We prove that a relative equilibrium formed of…
Consider the three-body problem with an attractive $1/r^2$ potential. Modulo symmetries, the dynamics of the bounded zero-angular momentum solutions is equivalent to a geodesic flow on the thrice-punctured sphere, or ``pair of pants''. The…
A new coordinate system is defined for the Four-Body dynamical problem with general masses, having as its origin of coordinates the center of mass. The transformation from the inertial coordinate system involves a combination of a rotation…
We construct a highly-symmetric periodic orbit of eight bodies in three dimensions. In this orbit, each body collides with its three nearest neighbors in a regular periodic fashion. Regularization of the collisions in the orbit is achieved…
The seminal work of Coleman, Glaser, and Martin established that, at zero temperature, any non-trivial solution to the equations of motion with the least Euclidean action is $O(D)$-symmetric. This paper extends their foundational analysis…
A model one-dimensional self consistent steady state collisionless self-gravitating system in which all the particles have the same energy is presented. This has the remarkable property that the position and velocity of the particles…
In a general setting of a Hamiltonian system with two degrees of freedom and assuming some properties for the undergoing potential, we study the dynamics close and tending to a singularity of the system which in models of $N$-body problems…
We study the efficiency of sliding locomotion for three-link bodies in the presence of dry (Coulomb) friction. Friction coefficient space can be partitioned into several regions, each with distinct types of efficient kinematics. These…
The subject is brake orbits for the 3-body problem: orbits where all velocities are zero at some instant. We extract a paradox and a mystery out of a recent database of 30 non-collision periodic brake orbits for the equal mass 3-body…
One of the outstanding problems of classical celestial mechanics was the restricted 3-body prob- lem, in which a planetoid of small mass is subject to the Newtonian attraction of two celestial bodies of large mass, as it occurs, for…
The article formulates the classical three-body problem in conformal-Euclidean space (Riemannian manifold), and its equivalence to the Newton three-body problem is mathematically rigorously proved. It is shown that a curved space with a…
The dynamics of a constrained three-vortex problem, a free point vortex pair in the velocity field of a fixed point vortex, is investigated. The underlying dynamical system is simplified using a coordinate transformation and categorized…
The Kepler-Heisenberg problem is that of determining the motion of a planet around a sun in the Heisenberg group, thought of as a three-dimensional sub-Riemannian manifold. The sub-Riemannian Hamiltonian provides the kinetic energy, and the…
We report on figure-eight choreographic solutions to a system of three identical particles interacting through a potential of Lennard-Jones type, $1/r^{12}-1/r^6$ where $r$ is a distance between the particles. By numerical search, we found…
In classical mechanics, the Kepler potential and the Harmonic potential share the following remarkable property: in either of these potentials, a bound test particle orbits with a radial period that is independent of its angular momentum.…
We consider the special case of the restricted circular three-body problem, when the two primaries are of equal mass, while the third body of negligible mass performs oscillations along a straight line perpendicular to the plane of the…
This paper is an extension of an earlier paper that dealt with global dynamics in autonomous triangular maps. In the current paper, we extend the results on global dynamics of autonomous triangular maps to periodic non-autonomous triangular…
In this paper, we study the planar circular restricted three-body problem for energy levels slightly above the first critical value. We first observe that the energy hypersurfaces in the Birkhoff regularization corresponding to these energy…